DYNAMICS OF A VECTOR-BORNE MODEL WITH DIRECT TRANSMISSION AND AGE OF INFECTION

. In this paper we the study of dynamics of time since infection structured vector born model with the direct transmission. We use standard incidence term to model the new infections. We analyze the corresponding system of partial diﬀerential equation and obtain an explicit formula for the basic reproduction number R 0 . The diseases-free equilibrium is locally and globally asymptotically stable whenever the basic reproduction number is less than one, R 0 < 1. Endemic equilibrium exists and is locally asymptotically stable when R 0 > 1. The disease will persist at the endemic equilibrium whenever the basic reproduction number is greater than one.

1. Introduction. Zika virus (ZIKV) is a flavivirus, transmitted by the Aedes aegypti mosquitoes as for the other vector borne diseases such as malaria, dengue fever, and West Nile virus. ZIKV was first isolated in the Zika forest in Uganda from the rhesus monkey in 1947 [13,21]. Outside Africa and Asia, first outbreak of Zika virus was reported in Yap State, part of the Federated States of Micronesia in 2007. After this outbreak, between 2007 − 2016, the spread of Zika virus infections have been reported around the world, including in southeast Asia; French Polynesia and other islands in the Pacific Ocean; and parts of South, Central, and North America [24,25,26].
The Zika virus infection causes mild or no symptoms [23]. However, Zika infection during pregnancy can cause serious birth defects and ZIKV infections were found to be connected with Guillain-Barre syndrome and Microcephaly [27,45]. When a person develops Guillain-Barre, their body's immune system mistakenly attacks part of its peripheral nervous system whereas in Microcephaly, Zika affects the brain, causing swelling of the brain or spinal cord or a blood disorder which can result in bleeding, bruising or slow blood clotting [14,40,39]. To the date, there is no specific medicine or vaccine to prevent or to treat Zika disease [38]. So preventive measures is the most effective way to prevent the infection, especially to pregnant women [15,16].
The main route of transmission for Zika virus is through the bite of an infected mosquito, but Zika has also direct transmission: through sexual contact, vertical transmission or blood infusion [18,22,42,43,36]. Direct transmission of Zika has been documented in nine countries-Canada, Argentina, Chile, France, Italy, New Zealand, Peru, Portugal and the United States of America [37].
Several mathematical models have been used to understand the transmission dynamics of vector borne diseases [30,31,32,33,19,44,34]. Ordinary differential equation (ODE) models dealing with the ZIKV disease have been proposed and extensively analyzed in past years [42,17,28,34,29,41]. In this study, we present a mathematical model of ZIKV incorporating both vector and direct transmission where infected individuals are structured by time-since infection. Transmission and recovery varies during the infectious period. Hence infection age affects the number of secondary infected individuals [35]. In Section 2, we introduce a model of ZIKV with age of infection where vector and direct transmission are both modeled as standard incidences. Then, in Section 3, we study the local stability of the disease free and endemic equilibrium and determine the reproduction number R 0 . It is followed by Section 4, where we discuss global stability of the disease free equilibirum. In Section 5, we present the persistence of the endemic equilibrium.
2. Vector-borne model with direct transmission. To model the spread of Zika infection in a population, we divide total host population into three non intersecting classes: susceptible, infected and recovered individuals. Since infectivity for infectious individuals varies with time since infection, we structure the infected class by time since infection parameter τ . The time since infection begins when an individual becomes infected and progresses with the chronological time t. Let i(τ, t) be the density of infected individuals at time since infection τ and at time t, S(t) be the number susceptible individuals and R(t) be the number of recovered individuals at time t. Then N (t) denotes the total human population. Here, we use an endemic model since the Zika epidemic has continued for nearly three years. Researchers estimate the reproduction number of Zika virus infection to be greater than one, and suggest that the virus is endemic [12]. Susceptible individuals are recruited at a rate Λ and µ denotes the natural death rate for humans. The vector transmission and direct transmission are both modeled as standard incidence, given by where β v is the vector transmission rate and β d (τ ) is the direct transmission rate which depends on the time-since infection variable τ of the infected host. Zika infection with a given strain (lineage) is believed to offer long-term protection [20], so the model does not include the possibility of subsequent infections of recovered individuals. The recovery rate γ(τ ) is also assumed to depend on time-since-infection The vector population is divided into 2 non-intersecting classes, susceptible vectors, S v (t), and infected vectors, I v (t). Vectors do not clear infection, so we use an SItype model for their dynamics. Since mosquitoes feed on many other species, we assume that the abundance of vectors does not depend on the abundance of humans.
In the model, µ v is the natural death rate of vectors and Λ v is their recruitment rate per unit time. Human to vector transmission is also modeled as standard incidence, where β(τ ) is the transmission rate from infectious hosts (i(τ, t)) to vectors which depends on host's age of infection. Models of vector-borne diseases with direct transmission is not new and have been considered before as ODE models for homogeneous population [17]. Including direct transmission with the vector transmission, we obtain the following model structured with time-since-infection, Age of infection dependent transmission parameter-functions β(τ ) and β d (τ ) are bounded and uniformly continuous defined on a compact support with a non-zero Lebesgue measure. Recovery rate γ(τ ) belongs to L ∞ (0, ∞). The dependent variables and their meaning are listed in the Table 1. The parameters and their meaning in the models are listed in the Table 2.
The total host population is for t ≥ 0, which satisfies the following differential equation, The number of susceptible vectors at time t The number of infected vectors at time t S(t) The number of susceptible individuals at time t i(τ, t) Density of the infected host with infection age τ at time t R(t) The number of recovered individuals at time t µ v Therefore the following set Ω is positively invariant for the system (1).
where R + = [0, ∞) and Since the exit rate of infectious individuals is given by the term γ(τ ) + µ, the probability of still being infectious τ units of time after being infected is given by Then the reproduction number of the disease in system (1) is given by Epidemiologically, reproduction number gives the number of secondary infections produced by one infected individual in a totally susceptible population during its lifetime as infection. System (1) has three modes of infection, vector-to-human R v , human-to-vector R h and human-to-human R d transmission, and each given by Thus the reproduction number is Next, we use the approach first introduced by Thieme [2] and also adopted by the authors [1,8,9,10,11]. We define the semiflow U of the solutions of the system (1) as Let the state space U be defined as U = R × R × R × L 1 (0, ∞) × R, then for any u = (u 1 , u 2 , u 3 , u 4 (τ ), u 5 ) ∈ U its norm is given as We would like to express (1) as an ODE on a Banach space. First, we move the non-linearity in the boundary condition in (1) to a nonlinear operator by enlarging the state space U . Let X = U × R and That is, for any vector u = (u 1 , u 2 , u 3 , u 4 (τ ), u 5 ) ∈ U , u = (u 1 , u 2 , u 3 , u 4 (τ ), 0, u 5 ) ∈ X 0 . We denote the positive cone in the corresponding space by We define a linear operator A : D(A) ⊂ X 0 → X as Note that D(A) = X 0 , hence the linear operator A is not dense in X. Thus A can not be a generator of a C 0 -semigroup. Next, we define a nonlinear map F : X 0 → X as We now define the following semi-linear Cauchy problem Since A is not dense in X and not bounded, the abstract Cauchy problem (3) may not have strong solutions, since u(t) is neither differentiable nor an element in D(A).
In other words, if u 0 ∈ X 0 \ D(A), we may not obtain strong solutions of (3). An approach to fix this problem is obtained by integrating (3) in time, then we get A continuous solution to (4) is called an integral solution to (3). The linear operator A is not densely defined but it satisfies the estimates of Hille-Yosida Theorem as shown in Proposition 1. Letμ = min{µ v , µ} and ρ(A) denote the resolvent of A.
Furthermore, the non linear operator F (u) satisfies Lipschitz condition.
Proof. Let's denote F(u) as; . Note that the total population is bounded below away from zero, and we set N (u) > N u0 . Then With similar analysis, one can show that By applying the results in [2], we obtain the following result.
Theorem 2.1. The system (1) represented by the integral equation (4) has a unique continuous solution with values in X 0 + . Thus, there exists uniquely defined semiflow {U(t)} t≥0 on X 0 + such that for each u 0 ∈ X 0 + , there exist a continuous mapping This implies for sufficiently large λ, λ(λI − A) −1 maps X + into itself. Next, we 3. Steady states and their local stability. Next, we study equilibria of (3). First, we recall the following Theorem from [2].
Theorem 3.1. The following statements are equivalent for u * ∈ X 0 i. u(t) = u * is time indepentent solution to (4); ii. U(t)u * = u * for all t ≥ 0; iii.u * ∈ D(A) and Au * + F (u * ) = 0. i ii. When R 0 > 1, then there exist a unique endemic equilibrium given by Proof. Suppose u * ∈ X 0 is an equilibrium of (3), then Au * +F (u * ) = 0. The system for the equilibria takes the following form Clearly, i * (τ ) = i * (0)π(τ ). If i * (0) = 0, then we obtain disease free equilibrium Solving for I * v , we get, Solving first equation of (6) for S * v gives, and solving for S * , we get which can be written as quadratic equation Thus, there exists a unique positive i * (0) when R 0 ≥ 1. When R 0 < 1 then a 0 > 0 and Thus the endemic equilibrium only exists when R 0 ≥ 1.
First we will be interested in A 0 , the part of A in D(A) which is defined as Thus, Clearly, A 0 is an infinitesimal generator of a strongly continuous semigroup {T A0 (t)} t≥0 of a bounded linear operators in D(A). Furthermore, the semigroup {T A0 (t)} t≥0 generated by A 0 is a contraction. That is there exists M 4 ≥ 0 such that To continue with the analysis, we recall the following definitions from [2,1].
We define the essential growth bound ω 0,ess (A) ∈ [−∞, ∞] of A as, where · ess denotes an appropriate measure of noncompactness of an operator.
Note that the resolvent set of the operator (A + DF (E 0 )) 0 consists of λ ∈ C such that λv − (A + DF (E 0 )) 0 v = u has a unique solution v ∈ D(A) for any u ∈ D(A). That is, Solving the system (8), we obtain, Thus, the system (8) has a unique solution if and only if 1 = F (λ), where Next, we solve the characteristic equation F (λ) = 1 to determine the stability of the disease-free equilibrium E 0 . It is clear that for any real λ, lim λ→∞ F (λ) = ∞ and F (λ) < 0. Since F (0) = R 0 and R 0 < 1, the equation F (λ) = 1 has no positive real root. Now, suppose that all complex roots have non-negative real parts, that is λ = a + ib with a ≥ 0, then |F (λ)| ≤ R 0 < 1. Thus all roots of F (λ) = 1 have negative real parts when R 0 < 1. When R 0 > 1, since F (0) > 1, the equation F (λ) = 1 has at least one positive solution. That is the disease-free equilibrium E 0 is locally asymptotically stable when R 0 < 1 and unstable when R 0 > 1.

Proof. The linearized equation of (3) around the endemic equilibrium
Following the steps in Theorem 3.4, by Corollary 4.3 in [2], local stability is determined by point spectrum of (A + DF (E * )) 0 . Thus, setting Requiring v 2 = 0 and v 4 (0) = 0 gives the characteristic equation as F (λ) = G(λ) where Let λ = a + ib with a > 0. Clearly |F (λ)| > 1 and Thus, any λ with positive real parts cannot satisfy the equation F (λ) = G(λ). Therefore, the endemic equilibrium is locally asymptotically stable in this case.
4. Global Stability of Disease Free Equilibrium. We next establish the global stability of the disease-free equilibrium E 0 . In showing the global stability of disease free equilibrium, we need following Fluctiation Lemma (Lemma 4.1) and Lemma 4.2.
Theorem 4.3. When R 0 < 1, the disease-free equilibrium E 0 is globally asymptotically stable.
Proof. Integrating the infected class i(τ, t) along the characteristic lines, we obtain where Using Lemma 4.2, and Integrating the infected vector class, and Substituting (9) and applying Lemma 4.2 gives lim sup Thus Substituting (10) and taking lim sup yields which gives lim sup Next, we show that lim By Lemma 4.2 we obtain, Similar results are true for bounded functions β d (τ ) and γ(τ ). Next, we show that lim It follows from . Similar argument shows lim t→∞ R(t) = 0.

5.
Persistence. When R 0 > 1, the disease free equilibrium is unstable. We will show that, when R 0 > 1 the system (1) is persistent and hence the disease will establish. That is we need to first make sure that the initial conditions are nontrivial and they lead to new infections. To be precise, suppose that Notice that the space X 0 + can be identified with the space M 0 = R + × R + × R + × L 1 + (0, ∞)×R + . As stated in (2), M 0 is positively invariant under the semiflow U(t). Furthermore, letM = R + × R + × R + ×M × R + , then set M = M 0 ∩M Let ρ i : M → R + for i = 1, 2, 3 be defined as follows, Proposition 3. Let R 0 > 1, then the system (1) is uniformly weakly persistent.
That is there exists > 0 such that, Proof. We will argue by contradiction. Assume that for each > 0, then either one of the inequalities will have lim sup Hence there exists T > 0 such that for each t > T , we have Withous loss of generality, we can assume that above inequality hods true for all t ≥ 0, (that is T = 0) by shifting the dynamical system. From the equations involving susceptible vectors and susceptible hosts in (1), we obtain And similarly for the infected vector class we have, We next take the Laplace transform of both B(t) and I v (t). We denote the Laplace transforms of B(t) and I v (t) asB(λ) andÎ v (λ) respectively. We obtain, I v (λ) from the sytem above, we obtain which gives a contradiction.
A consequence of the system (1) being uniformly weakly persistent is that the disease-free equilibrium is unstable. We proceed by showing that the semiflow U(t) : [0, ∞) × M → M of the system (1) [3]) For each t > 0 suppose U(t) =Ũ(t)+Û(t), wherẽ U is completely continuous. Suppose there exists a function k : R + × R + → R + such that k(t, r) → 0 as t → 0 and Û (t)u 0 ≤ k(t, r). Then U(t) is asymptotically smooth. First, we show that the semiflow U(t) is asymptotically smooth. Proof. We will apply Lemma 5.2 to show that U(t) is asymptotically smooth. We begin by splitting the semiflow into two compartments, U(t) =Ũ(t)+Û(t) as follows: Note that S v (t), I v (t), S(t) and R(t) satisfy the sytem (1) with i(τ, t). Since, both i andî are non-negative,Ũ ≤ U andÛ ≤ U. Note that, Hence, Û (t)u 0 → 0 as t → ∞. Next, we show thatŨ(t) is completely continuous. A semiflowŨ(t) is called completely continuous if for each fixed t, the family of functions K t = {Ũ(t)u 0 : u 0 ∈ B} is precompact for a bounded set B contained in M (p.36 [3]). So, assume that the initial conditions are in a abounded set B, that is u 0 ∈ B, such that u 0 = |S v0 | + |I v0 | + |S 0 | + i 0 L 1 + |R 0 | ≤ r for some constant r. By (2), we have K t = {Ũ(t)u 0 = (S v (t), I v (t), S(t),ĩ(·, t), R(t))} ⊂ Ω and hence bounded. We will show that the family of functions, K = {ĩ(·, t) :Ũ(t)u 0 ∈ K t } for any fixed t is precompact by Frechet-Kolmogorov Theorem [4]. Frechet-Kolmogorov Theorem states that K ⊂ L 1  (11): Thus, we first need to show that both B(t) and B (t) are bounded.
On the other hand, Hence |B (t)| ≤ k 3 + k 4 t and similarly, |B (t)| ≤k 2 . Then, Next, we show that the semiflow U(t) has a global compact attractor.
Proposition 5. When R 0 > 1, then the semiflow U(t) has a global compact attractor.
Proof. We will apply Lemma 5.3. Thus, we need to show that U(t) is point dissipative and orbits of bounded sets are bounded. A semiflow U(t) is called point dissipative, if there exists a bounded set B ⊂ M that attracts each point of M [3]. Clearly by (2), the semiflow U(t) is point dissipative. Again by (2), orbits of bounded sets are bounded. That is U(t)u 0 ≤ Λ µ + Λ v µ v for all t ≥ 0 whenever u 0 ≤ c 1 for some constant c 1 . Thus semiflow U(t) has a global compact attractor.
We continue by showing that the disease is uniformly strongly persistent. We will prove that the vector-borne disease in system (1) is uniformly strongly ρ-persistent, by applying the Theorem 2.6 in [5]. The Theorem 2.6 in [5] states the following.
Lemma 5.5. (Theorem 2.6 in [5]) Let U(t) be a continuous semiflow on a metric space M which has a compact attractor A. Furthermore, we assume for any total orbit φ(t) : R → M, if s ∈ R and ρ(φ(s)) > 0, then ρ(φ(t)) > 0 for all t > s. Then U(t) is strongly ρ-persistent whenever it is uniformly weakly persistent. Proposition 6. Let R 0 > 1, then the system (1) is uniformly strongly ρ-persistent. That is there exists > 0 such that, Proof. We will apply Theorem 2.6 in [5]. Total orbits are the solutions of the system (1) defined for all time t ∈ R. By (2), the solutions of the semiflow is non-negative. Hence, integrating the differential equation of I v , we obtain I v (t) ≥ I v (s)e −µv(t−s) for all t > s. Therefore, Similarly, Thus, ρ(φ(t)) > 0 for all t > s, whenever I v (s) > 0. So, by Theorem 2.6 in [5], semiflow U(t) is uniformly strongly ρ-persistent, whenever it is uniformly weaklypersistent, that is when R 0 > 1. 6. Conclusion. Infection age plays a vital role in the transmission of ZIKV. In this study, we formulate a hyperbolic PDE model of Zika virus infections, which includes both vector and direct transmissions and where the new infections are modeled as standard incidence. We obtain the explicit representation of the reproduction number, R 0 . We showed that the disease-free-equilibria is locally and globally stable when R 0 < 1. We also showed endemic equilibrium is locally stable when R 0 > 1. Persistence of endemic equilibrium is established when R 0 > 1, however global stability of endemic equilibrium for R 0 > 1 is still an open question.